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# Four points that form a polygon lie on the circumference of

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Intern
Joined: 06 Jun 2010
Posts: 24
Four points that form a polygon lie on the circumference of  [#permalink]

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12 Mar 2012, 07:07
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Difficulty:

5% (low)

Question Stats:

85% (00:50) correct 15% (00:54) wrong based on 92 sessions

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Four points that form a polygon lie on the circumference of the circle. What is the area of the polygon ABCD

(1) The radius of the circle is $$\frac{\sqrt{2}}{2}$$
(2) ABCD is square.

The figure is drawn in such a way that the two angles are shown to be right angles.

Kaplan book says both statements are required to solve, but i think that state 1 alone is sufficient.

Given that we know the angle is a right angle and that a diagonal AC is joined as well. We can say that AC is a diameter (angle inscribed in a semi circle). And if we join point D with say centre O, OA OD And OC will be equal. We already know the radius. OD will be perpendicular to the base (diameter AC) and hence we can get the area. We dont need to know whether ABCD is square or not.

Could someone shed some light ?

Edit: Apologies for amateur fine art skills

Thanks

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Untitled.png [ 3.68 KiB | Viewed 2323 times ]

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Joined: 02 Sep 2009
Posts: 47983
Re: Finding area of a polygon inscribed in a circle  [#permalink]

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12 Mar 2012, 09:26
1
1
mist3rh wrote:
Source: Kaplan

Four points that form a poylgon lie on the circumference of the circle. What is the area of the polygon ABCD

1.) the radius of the circle is \sqrt{2}/2
2.) ABCD is square.

The figure is drawn in such a way that the two angles are shown to be right angles.

Kaplan book says both statements are required to solve, but i think that state 1 alone is sufficient.

Given that we know the angle is a right angle and that a diagonal AC is joined as well. We can say that AC is a diameter (angle inscribed in a semi circle). And if we join point D with say centre O, OA OD And OC will be equal. We already know the radius. OD will be perpendicular to the base (diameter AC) and hence we can get the area. We dont need to know whether ABCD is square or not.

Could someone shed some light ?

Edit: Apologies for amateur fine art skills

Thanks

The red part is not correct.

Four points that form a polygon lie on the circumference of the circle. What is the area of the polygon ABCD

(1) The radius of the circle is $$\frac{\sqrt{2}}{2}$$ --> consider two quadrialaterals below:
Attachment:

Inscibed polygon.png [ 40.54 KiB | Viewed 2307 times ]
Do they have the same area? No. So this statement is not sufficient.

(2) ABCD is square. We know nothing about the measurements of either the square or the circle. Not sufficient.

(1)+(2) We know the radius of a circle and that ABCD is a square, which is sufficient to find the area.

Hope it's clear.
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Re: Four points that form a polygon lie on the circumference of  [#permalink]

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12 Mar 2012, 10:56
Thanks mate ! The problem shows the diagram in a way that reflects that point D joined with centre O would be perpendicular (given that diagrams are drawn to scale unless specified). But obviously thats not the case. The diagram positioning was primarily why i assumed OD would be perpendicular. I hope GMAT does not have such ambiguities.
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Re: Four points that form a polygon lie on the circumference of  [#permalink]

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12 Jul 2018, 11:02
Hello from the GMAT Club BumpBot!

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Re: Four points that form a polygon lie on the circumference of &nbs [#permalink] 12 Jul 2018, 11:02
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